I need help making a non realistic spider web so I can 3D print tokens for a game
You can use the Proportionnal Editing Tool for that. It's a tool that allow to move only certain vertices, but the surounding vertices are kinda attracted to them.

Select your ring and enter Edit mode (Tab). Select the vertices near every intersection. You can use the Circle select ("C") to do it easily.


Press the "O" key to activate the PET, and select a type of interpolation in this list (Sharp should give good results) 


Now scale the vertices up with the "S" key. While you are scaling, you can adjust the effect radius of the PET whith the mouse wheel.